In this work, we consider fracture propagation in nearly incompressible and (fully) incompressible materials using a phase-field formulation. We use a mixed form of the elasticity equation to overcome volume locking effects and develop a robust, nonlinear and linear solver scheme and preconditioner for the resulting system. The coupled variational inequality system, which is solved monolithically, consists of three unknowns: displacements, pressure, and phase-field. Nonlinearities due to coupling, constitutive laws, and crack irreversibility are solved using a combined Newton algorithm for the nonlinearities in the partial differential equation and employing a primal-dual active set strategy for the crack irreverrsibility constraint. The linear system in each Newton step is solved iteratively with a flexible generalized minimal residual method (GMRES). The key contribution of this work is the development of a problem-specific preconditioner that leverages the saddle-point structure of the displacement and pressure variable. Four numerical examples in pure solids and pressure-driven fractures are conducted on uniformly and locally refined meshes to investigate the robustness of the solver concerning the Poisson ratio as well as the discretization and regularization parameters.
翻译:在这项工作中,我们考虑使用一个阶段式配方,在几乎不压缩和(完全)不压缩的材料中进行骨折扩散;我们使用一种混合形式的弹性方程式,以克服体积锁定效应,并为由此形成的系统制定一个稳健、非线性和线性求解器计划和先决条件;同时,由单一的单一解决的不平等系统,由三种未知因素组成:流离失所、压力和相位场;由于混合、成文法和裂痕不可逆转性造成的非线性,通过部分差异方程式中非线性牛顿综合算法解决了非线性;我们使用一种初步的硬性活性战略克服了体积的锁定效应,并为裂变性制约制定了一种原始的主动战略;每个牛顿步骤的线性系统以灵活、通用的最低限度剩余方法(GMRES)反复解决;这项工作的主要贡献是开发一个针对具体问题的先决条件,利用流离失所和压力变量的马鞍点结构;四个纯固体和压力驱动的断裂变形的数值示例是统一和当地改良的模件,以调查关于离子和离心率比率参数的固化参数。